#152175

order	512
nilpotency class	5
isoclinism	isoclinic class of #13669
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#3417	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

612 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,11,17,13,8,10,16)(7,19,23,50,25,51,22,49)(9,28,32,63,34,60,31,62)(12,36,35,27,38,47,30,18)(14,39,41,54,20,52,40,21)(15,43,46,67,26,58,45,33)(24,56,37,44,57,68,55,59)(29,48,65,42,66,61,64,53);(5,7)(6,14)(8,20)(9,29)(10,22)(11,23)(12,37)(13,25)(15,44)(16,40)(17,41)(18,48)(19,28)(21,33)(24,30)(26,59)(27,61)(31,64)(32,65)(34,66)(35,57)(36,42)(38,55)(39,43)(45,56)(46,68)(47,53)(49,62)(50,63)(51,60)(52,58)(54,67);(2,4)(5,8,13,6)(7,20,25,14)(9,26,34,15)(10,16,11,17)(12,27,38,18)(19,21,51,54)(22,40,23,41)(24,53,57,42)(28,33,60,67)(29,59,66,44)(30,47,35,36)(31,45,32,46)(37,61,55,48)(39,50,52,49)(43,63,58,62)(56,65,68,64)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=gh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dg,g^b=gh,h^b=hi,e^c=ei,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59371);